"if you think childlike, you'll stay young. If you keep your energy going, and do everything with a little flair, you're gunna stay young. But most people do things without energy, and they atrophy their mind as well as their body. you have to think young, you have to laugh a lot, and you have to have good feelings for everyone in the world, because if you don't, it's going to come inside, your own poison, and it's over" Jerry Lewis
"I don’t believe
in the irreversibility of situations" Deleuze

Note on Citations

The numerical citations refer to page number. The source's text-space (including footnote region) is divided into four equal portions, a, b, c, d. If the citation is found in one such section, then for example it would be cited p.15c. If the cited text lies at a boundary, then it would be for example p.16cd. If it spans from one section to another, it is rendered either for example p.15a.d or p.15a-d. If it goes from a 'd' section and/or arrives at an 'a' section, the letters are omitted: p.15-16.

I came across the following video (through Deleuze scholar Rockwell Clancy’s facebook feed). If you have not seen it, I think you might be amazed for reasons that could be further examined. [Best seen on youtube: http://www.youtube.com/watch?v=FTByHbjgz8k#t=27]

The following is highly experimental and is meant to serve as the starting mistakes for a larger project currently under development.

This project examines Gilles Deleuze’s ‘Logics’. We have two “logic” books by Deleuze, Francis Bacon: The Logic of Sensation and The Logic of Sense. I believe that Deleuze’s Cinema 1 and 2 comprise a third logic book, what we might title The Logic of Signs. All three ‘logics’ share the common logic of affirmative synthetic disjunction [I discuss this further pp.204-205 of "In the Still of the Moment"]. Informally, in Deleuze’s logic, incompatible states of affairs are given as forced together. There are 'tensions' between them on account of their contrariety. More formally speaking, say that term B is not term A. Now note that when A is conjoined with not-A, we have a contradiction. But there is no ‘sense’ or meaning resulting when A and A are conjoined self-redundantly. However, there is significance when information is non-redundant with itself, and thus introduces contradictory combinations. We find in the world both A and B at once (in contradictory states of affairs between one instant and the next, for example) which means we find A and not-A in combination. We have a contradiction that is real, true and meaningful [and such a logic of contradiction is useful for accounting for change and becoming]. I mean meaningful in a number of ways:

In the Logic of Signs / the Logic of Sensibility: Synthesis of Contradictory Structures[What are the basic structures that make relations sensible?]

Prelinguistic, signaliticmeaning [in the sense of pre-linguistic signs. See Deleuze Cinema 2, chapter 2. For this, we examine structures that create the pathways or tendencies by which specific terms will relate in meaningful ways].

There are basic structures that relate series of terms under certain modes of conjunction. For example, Deleuze shows how perceived variations of depth in film can be conjoined with variations of time in the story. The co-presence of different spatial points then implicitly suggests the contradictory co-presence of temporally distinct moments of time. More specifically, the temporal relations ‘before’ and ‘after’ are given in depth of field shots. Let's look quickly at some relevant scenes in Orson Welles' Citizen Kane. The film is largely composed of flashbacks from people who knew Kane. In one case it is his second wife, who enjoyed singing but never aspired to be a professional, and she was never gifted enough to be one in the first place. When she is being interviewed about Kane's past, the camera as you can see in the clip below starts from high and seemingly moves down onto the wife, as if we are falling through the depths of time by probing into her memories of the distant past.

So because the camera motion and flashback editing are moving us an extent similarly through both space and time, we have already
a structuring principle of visual depth equaling temporal depth. What we then see first is Kane's wife being trained for professional opera singing. Despite her hopeless failings and unwillingness to continue, Kane is maniacally driven to make her a star, partly by using the influence of his newspaper to promote her. We will see a frantic montage with superpositions of intense imagery and with music that will build to a climax, then ending in darkness and silence. This sequence is a portrayal of the harsh intensity of the flow of time during her period of rehearsal and performance, ending in her suicidal breakdown. So first we are given an impression of time in its thickness as an intense and overfilled continuous flow. All the while, time is taking its toll on the wife, wearing her down and breaking her body and spirit, seemingly to her demise. Then we see Kane burst in on her, rescuing her from death. For Deleuze, this scene is important in its juxtaposition to the prior linear sequence. In the suicide scene, Kane stands at a distance from his wife, with a gulf of visual depth between them. The wife looks decrepit, showing the signs of the time that all the while Kane had been ignoring in his mania. Kane must face the time that has passed; he sees those hectic months all at once, at a distance, in their purified empty form. But as we said, visual depth already had a temporal meaning. We make sense of this scene because the depth tells us that Kane must face the period of time that he was destroying his wife, and cross through it rather than ignore it.

[If the above does not work, try this alternative:]

We might at this point note the connection Deleuze makes between Peirce's notion of the icon and the idea of 'analogy by isomorphism'. [My discussion of that connection is here.] We have two structures each of their own domain, namely, the structure of spatial extension and the structure of temporal extension. We might say that visual depth is an iconic presentation of temporal depth, as there is an isomorphic (one-to-one) relation between variations in distance in the one domain to the variations in time in the other. It is for this reason that visual depth is a 'sign' for temporal depth, with that temporal depth then providing a new series of relations to import into the spatial visual domain so to open an additional layer of meaning in the imagery and story line. Returning to our example, Kane does recognize the damage he did to his wife by exposing her to so much that harmed her. But his response is no better, and still shows a profound insensitivity to her needs. Previously she was flooded with damaging activity, with critical people interjecting in her life and breaking her morale. Afterward he does the opposite, still to her detriment. He isolates her, giving her too much space, and in a sense trying to create a protective 'empty' period in her life, like the vast deep emptiness of Kane's palace where she is locked up and socially isolated. In a sense, Kane does not really change as a person, despite the warning signs saying that his obsessively controlling character is damaging to the person in his life that he loves the most. In the end his wife leaves him. Kane then trashes her bedroom, walks through a hallway of mirrors, and sometime later dies saying 'Rosebud'. Rosebud is the name of his sled, and it serves to mark his transition as young child when he suddenly goes from poverty to immense inherited wealth. With the idea of visual depth and time already established in our minds, we then can use that schema for understanding that final mirror scene. So temporal variation can be mapped onto visual depth. In the mirror scene, Kane is projected infinitely into the vast depth between the mirrors. But he is the same, a repetition of a unvarying character. His development halted when he obtained his wealth, and he has tragically remained unchanged throughout the depth of his life. [In the clip below, the mirror scene is recalled by another witness.]

Logic of Sensibility in Volinez' Spears Video

Before
drawing any inferences about what we see, and before the sense data can
come into any additional relations, we first notice a basic structural feature in the video.
There is often a lot of visual depth in the original video, now forming a background to the flat plane
inserted on its surface. The new frame is sized and composed so to
appear as though it occupies one of the levels of depth in the broader image, looking like a movie screen standing up some distance into the field of visual depth of the scene.

There are even instances where he inserts his image onto mirrors to replicate a reflection.

The basic content of the overlaid image is often made so that it seems to extend past its boundaries into the background.

Here are some notable moments where the actions in the box are coordinated with the extremities of Spears' motions.

There is also a scene where he is made to seem as though his two-dimensional image moves through the three-dimensionality of the background. I include it with the original for comparison.

There is also a scene where he adjusts the color of the inner frame to match the background.

There also seem to be shots that would be too problematic to replicate, perhaps because of a difficult high angle. In these cases and in some others, he places a red dot over top of Spears. But to make that technique more seamless, he at times puts a red dot over his own face, perhaps only to equalize the instances.

However, despite these efforts to maintain the junction of the two series of moving images, there is still a strong tension between them. No matter how matched the images are, it still appears as though the overlaid image is two-dimensional and its surroundings three-dimensional. In some cases the match is noticeably off (perhaps intentionally), and in other cases the inserted image extends outside the frame of the background, reinforcing its two-dimensional overlaid look.

So while our eyes are forced to place the two series of images together, they are strongly incompatible. They are a cross-over of two different worlds, a two-dimensional one and a three-dimensional one. This is the basic structural feature that will create the basis for how the two series of terms relate differentially and meaningfully. In other words, this structure combines distinct series from separate domains, and by forcing them together, provides the conditions for the sensibility of their differential relations.

In the Logic of Sensation / the Logic of Sensitivity:

Synthesis of Contradictory Sense Data[How do the structures of relation bring together contradictory sense-data in an informative way?]

Affective meaning: Significant sense data. Our five senses provide us with data about the world around us (and within us). Yet, we do not find meaning in redundancies in sense information. In fact, our nervous systems 'desensitize' themselves to constancies of sensation [see Marieb and Hoehn, Bateson, and Bergson]. We are more sensitive it seems to differences and incompatibilities in sense data when the information does not ‘compute’ and calls for our closer attention. Consider being in a warm room during a cold winter day. After a while, we get cozy, and we begin to sense things other than the room's temperature. But when we go outside into the frigid cold, we are instantly very aware of the change in temperature. It is a difference that makes a difference. It tells us to change our behavior, to cover our exposed skin and hurry to our destination. So affective meaning is sense data that ‘tells’ us something even before we consciously interpret it. And the data here are not just for example the warmth of the room and the cold of the outside air; rather, the experience of the difference between them is itself the significant datum, the difference that makes a difference. [See this entry for more on Bateson's definition of information as 'difference that makes a difference.']To analyze how the structural features of the image bring about contradictory sensations, let's take an example from Deleuze's Logic of Sensation, the painting Figure at a Washbasin, 1976 by Francis Bacon.

Often times in Bacon's paintings, there is a shape enclosing a figure. This is a structural feature that organizes the 'forces' in the painting. In some cases, the forces are acting dually against one another, and in certain instances they might give the impression of interchanging in-and-out flows. In this painting, the circle may seem to be closing in on the body, squeezing it. But the figure then pushes outward on the circle, as it seems to be flexing as though resisting and pushing back on that pressure. And also, he seems to be evacuating and escaping the confines of the circle through the drain.

The structural feature of the painting, the enclosing circle, combines incompatible forces, namely those pressing in and those pressing out. These forces intersect and collide in the figure's body, making it shake and spasm. We have the sensation then of a motion over and above the simpler two. We have two unidiretional motions, and the third non-directional vibrational motion which is a disjunctive synthesis of the other two. [For more on Deleuze's analysis of the diastole/systole rhythm in this painting, look toward the end of this entry.]

The Logic of Sensitivity in the video

In the music video, we are also given this impression of a back-and-forth dance between the overlaid frame and the background. The box at some times tightens around the inserted dancer, while at other times he seems to push the boundaries outward. We do not get the impression of spasms in the video like we do in the painting. What we have instead it seems is a more erotic play of encroachment and retreat, and this comes not from the content of either series but in the interaction between them. Here are some instances where we see the box's boundaries in motion.

In the Logic of Sense / the Logic of Explanation:

Synthesis of Contradictory Inferences[We draw inferences from the given data. How do contradictions between series of inferences themselves have inferential value?]

Dramatic/literary/explanatory meaning. [Warning: this portion is problematic and vague.] A story is a series of rabbits out of hats, by which I mean, the events unfold without scientific predictability. If we could deduce the whole tale all the way down to its conclusion only from first hearing the beginning lines, then we would not need to follow along with it. Narrative events in a way are meaningfully connected but not logically implicit in one another. The tension between one trend in a story and a new divergent line beginning after a sudden twist has dramatic power to it. Consider an Aesop fable, 'The Fox and the Grapes':

A hungry Fox saw some fine bunches of Grapes hanging from a vine that was trained along a high trellis, and did his best to reach them by jumping as high as he could into the air. But it was all in vain, for they were just out of reach: so he gave up trying, and walked away with an air of dignity and unconcern, remarking, "I thought those Grapes were ripe, but I see now they are quite sour." [from Project Gutenberg]

Perhaps one way we mentally obtain explanations is by finding significance in differences between series of inferences in the story. In our example, we seem to have two pairings of inferences, with one set preceding the fox's change of mind, and another following that shift.Inference series A:The grapes are desirable. The fox is determined to eat them.Inference series B:The grapes are not desirable.The fox is not so determined to eat them.The difference and tensions between these series make sense if we add the dramatic moment. There is an aleatory point, a moment of uncertainty when the meanings shift [for more on narrative bifurcation, see pp.211-218 of "Still of the Moment"]. We might even say it is a moment of self-forgery [see this entry on the topic of self-development and the power of falsity]. The fox pretends to be the same self, but really he has changed, going from a determined creature to a less ambitious one. But he externalizes this change by revising the inferences of his world though his modifying the value of the grapes.Sadly I do not have a stronger methodology than this, but let's still experiment with it in the video. The main idea again is to look for inferential series and more importantly the contradictions between them, and asking what is the added significance of those contradictions.

Logic of Sense in the music video

[The warning continues to apply in the following.] Here we do not have a linear story. But we do have the inferred information from two series, that from in the box and that from outside it. We sometimes catch brief glimpses of Spears' obscured erotic body-presentation and movements. The background dancers reflect that, even when she is hidden. We can tell she has a slender and curved body. We might be led to draw certain inferences from this about female sexuality, for example, that it can be expressed by moving in a certain seductive way and by having a particular body-shape. Yet, even within the visible box the male dancer makes motions, gestures, and facial expressions that seem still somehow entirely fitting with this conventional view of feminine sexuality. In fact, we might even find his performance even more passionate in that regard. Here are some comparisons of particular expressions. You might find that Spears' movements seem more forced and mechanical.

So in the overlaid video, there is a tension between the series. Series A (of the background) leads to the inferences the main dancer is a slender woman moving seductively. Series B (of the overlaid foreground) makes us directly infer that the central figure is a large man also moving seductively. There is a series of tensions between the unfolding of these inferences, and the series combine all while strongly insisting on their incompatibility and contrariety. That tension could lead to inferences not found within either series, for example, possibly that there is no strong basis to distinguish male and female sexuality in the way that music videos might normally suggest, and it also might call into question the normal standards for the sexuality of women's appearance and self-presentation. The video is sexier with the large man, because he expresses eroticism more effectively with his more natural movements and facial expressions. This may not be the most interesting way to interpret the differential tensions between the two series of inferences. I chose it because it seemed the most obvious.

[The following is summary. My own comments are in brackets, but please consult the original text, as I am not a logician. All boldface and underlining are my own. Proofreading is incomplete so mistakes are still present.]

Graham Priest

In Contradiction: A Study of the Transconsistent

Part III. Applications

Ch.12. The Metaphysics of Change II: Motion

12.4 … And Its Consequences

Brief Summary:

Priest’s dialetheic Hegelean account of motion solves many of the problems created by the Russellean orthodox (‘at-at’) account of motion. For example, the orthodox account says that motion is made up only of states of rest, which is counter-intuitive. The Hegelean account however allows us to say that the object is both in a location and not in a location at the same time, and thus always is in a state of motion. It would even seem that time itself is structurally self-contradictory. For a dialetheic account, this does not mean that it is therefore non-real. Rather, it allows time to be both inconsistent and real.

The Hegelean state description of a body in motion, with its notion of the spread of locations at any time, makes quite precise Hegel’s claim that to be in motion is | to occupy more than one place (in fact a continuum of places) at the same time, and hence both to be and not to be in some place. It therefore renders quite rigorous his account of change. Moreover, the important defect of the account that I mentioned at the start of the last section, namely that it is unclear how the account relates to the canonical mathematical representation of motion, is clearly overcome. An equation of motion, x=f(t), still captures the idea that at time t the object is at f(t). It is just that there is more to change than this. It might be elsewhere too! [179-180]

Priest’s Hegelean account solved some of the problems he found with the orthodox account. Recall for example that it implies motion is constituted only by states of rest, and it is never actually in a state of motion. This seemed counter-intuitive. The Hegelean account allow for a moving body to occupy multiple locations for a single time point, so it does not have this problem. [180]

Also recall that in Zeno’s paradox of the arrow, the arrow was said to be in only one position at one time. Given the spread hypothesis, we can have the object in multiple locations for one time point. [p180]

Some things still need to be explored regarding the spread hypothesis. Nonetheless, we know it is preferable to the Russellean account. [180] Priest also notes that quantum indeterminacy might be explained using the spread hypothesis [for details see 180-181]

In fact, we might even say not only are objects in motion in two places at the same instant, but we might also say that time itself is structured as self-contradictory, with one moment occupying multiple time-points.

Let me end this chapter with one final application of the Hegelean account of change, where the change in question this time is not motion. Take any point of time, say, midnight on 1/1/2000. Then at this time ‘It is midnight on 1/1/2000’ is true. For a continuous period before and up to this time ‘It is not midnight on 1/1/2000’ is true. Hence by the LCC, this is true at midnight too. Thus, at this time it is both midnight on 1/1/2000 and not midnight on 1/1/2000. This application of the LCC is somewhat moot. It is not completely clear that ‘It is midnight’ and similar temporal claims describe states of affairs in the required sense of the word. But assuming that they do, the fact that such contradictions are produced, together with the Hegelean account of change, gives an exact and plausible sense to the obviously true and non-trivial claim that time itself is in a state of change or flux. This commonsense view has given all sorts of problems to the Russellean account of change. For, on the orthodox account, the view that time is itself in a state of change amounts to the banality that at one time it is one time, and at another, another. This has prompted a variety of responses of varying degrees of incredibility, from the view that time is not in a state of flux, to the view that there are ‘‘hypertimes’’. The contradiction theory of change solves the problem cleanly and swiftly. [181]

[Priest then raises the question of whether the spread hypothesis applies to either or both time understood as indexical temporal ‘A series’ and as non-indexical temporal ‘B series’. For details, see p.181.]

Some philosophers have concluded that time itself is inconsistent. But they further conclude that this means time is not real. Dialetheic logic, however, allows time to be both inconsistent and real.

A number of people have argued that time in itself is inconsistent. Many of these, such as the idealists Bradley and McTaggart, thought that for this reason it should be consigned to the realm of appearances, or of non-existence—though exactly what this means is not so clear. Dialetheism allows time to be both inconsistent and real. [181]

[The following is summary. My own comments are in brackets, but please consult the original text, as I am not a logician. All boldface and underlining are my own. Proofreading is incomplete so mistakes are still present.]

Graham Priest

In Contradiction: A Study of the Transconsistent

Part III. Applications

Ch.12. The Metaphysics of Change II: Motion

12.3 The Hegelean Account of Motion

Brief Summary:

In the Hegelean account of motion, we would think of there being a spread of moments to which the object occupies a spread of spaces. This duration is quite small and tight around a certain time point. Hegel’s view is that the position of the moving object is indiscernable at some moment, and so could be at one of many places in the same tiny moment.

Summary

Previously Priest examined the orthodox, Russellean, ‘at-at’, cinematic account of change. We found that it leads to the strange conclusion that motion is comprised of no more than states of rest. Now Priest will examine alternate accounts of motion, in particular Hegel’s.

[M]otion itself is contradiction’s immediate existence. Something moves not because at one moment of time it is here and at another there, but because at one and the same moment it is here and not here . . . [Hegel (1840), vol. 1, ch. 1, sect. C4., quoted in Priest 175]

Hegel means that although an moving object will be at different places at different times, it is necessary as well that at specific times it be in different places.

Hegel is not denying that if something is in motion it will be in different places at different times. Rather, the point is that this is not sufficient for it to be in motion. It would not distinguish it, for example, from a body occupying different places at different times, but at rest at each of these instants. What is required for it to be in motion at a certain time is for it both to occupy and not to occupy a certain place at that time. [175]

This account has not been well received, because it defines motion by means of contradiction. It is also not clear how exactly to relate this account with our more scientific and mathematical methods for calculating motion. However, the orthodox account of motion seems to be built into the formulas of calculus.

Thus, an equation of motion, x = f(t), just seems to encode the idea of the occupation of different places at different times: it merely records the correlation. By contrast, Hegel’s view seems to have no bearing on the matter. [176]

[Hegel reasoning for this seems to be that the object is at a single position at a single time, but near it are positions and times so close that we are unable to localize the body.]

The reason is roughly as follows. Consider a body in motion—say, a point particle. At a certain instant of time, t, it occupies a certain point of space, x, and, since it is there, it is not anywhere else. But now consider a time very, very close to t, t'. Let us suppose that over such small intervals of time as that between t and t' it is impossible to localise a body. Thus, the body is equally at the place it occupies at t', x' (≠x). Hence, at this instant the body is both at x and at x' and, equally, not at either. This is essentially why Hegel thought that motion realises a contradiction. [176]

Hegel also explains why we cannot localize the positions in an instant. It is because they fall along a continuum, and neighboring points along a continuum merge.

Hegel gives a reason why a moving body cannot be localised. The reason derives from his view of the continuum. Essentially, it is that in a continuum distinct points themselves merge. Thus, the reason why we cannot localise a body to t is just that t itself is not ‘‘localisable’’. As he puts it, {quoting Hegel (1940)}

[W]hen . . . we admit that time and space are continuous, so that two periods of time or points of space are related to one another as continuous, they are, while being two, not two, but identical . . . [M]ovement means to be in this place and not to be in it, and thus to be in both alike; this is the continuity of space and time which first make motion possible. {end quote}

And again: {quoting Hegel (1930)}

[When a body is moving] there are three different places: the present place, the place about to be occupied and the place that has just been vacated; the vanishing of the dimension of time is paralysed. But at the same time there is only one place, a universal of | these places, which remains unchanged throughout all the changes; it is duration existing immediately in accordance with its Notion, and as such it is Motion. {Priest 176-177, quoting Hegel, firstly “Hegel (1840), vol. I, pp. 273, 273–4 of the translation;” and secondly “ Hegel (1830), p. 43 of the translation. The italics are original.”}

Hegel held some interesting ideas that we might want to further pursue, for example the 18th century notion of the variable point and the contradictory unity of the discrete and the continuous. But for now, Priest will formulate Hegel’s main insight as “the Spread Hypothesis”. [177]

Spread Hypothesis [heading is bold in Priest’s text]

A body cannot be localised to a point it is occupying at an instant of time, but only to those points it occupies in a small neighbourhood of that time. [177]

Although this might seem like a strange concept to use in physics, we already know that strange things happen at the scale of Planck’s constant. [177]

Priest will articulate the spread hypothesis using the tense logic semantics he previously described. [177]

[In the following, Priest will first formulate the Russellean at-at account, which holds that a moving body cannot be in two places at the same time. The formulations he will give are basically saying that if a moving object is found in its mathematically determined location at a specific time, then this is true, but if it is not there, it is false. Recall that v is the function that assigns truth/falsity values (0/1 values) to the given proposition (stating the object’s position). The body is called b. The proposition has the structure ‘b is at point x’, which can be expressed as the relation Bx. The function determining the objects position is x = f(t). We might read the two formulas as (1a) the statement ‘b is at point r’ is true (at time t) if r equals the value that the function produces for that given time. And, (1b): the statement ‘b is at point r’ is false (at time t) if r does not equal the value that the function produces for that given time.] The following formulation we will call the “Russellean state description”:

Now, consider a body, b, in motion. Again to keep things simple, let us suppose that it is moving along a one dimensional continuum, also represented by the real line. Let us write Bx for ‘b is at point x’. Let us also suppose that each real, r, has a name, r. This assumption is innocuous. It could be avoided by talking in terms of satisfaction rather than truth. I make it only to keep the discussion at the propositional level. Let the motion of b be represented by the equation x = (t). Then the evaluation, v, which corresponds to this motion according to the Russellean account, is just that given by the conditions:

[177]

Priest draws a diagram to depict it. As we can see, only the proper time/place coordinate for f(t) is true.

[In the next formulation Priest describes the Hegelean account of motion. It seems he is saying that we need to think not just of single time points but as well time points surrounding in a set of time points here called θt. It also seems to be saying that although there are different instants in this set which correspond to different locations, if these instants are included in set θt, which surrounds specific point t, then it is true that the object is in these other locations. Specifically the formulations might be read (2a) the proposition ‘b is at location r’ is true if within the spread of moments around t (that is, in set θt), there is at least one time point which when used in the function produces that value for r. And (2b) the proposition ‘b is at location r’ is false if within the spread of moments around t (that is, in set θt), there is at least one time point which when used in the function produces a value that is not r. So while θt might be the set of time points around t, there is also the resulting ‘spread’ of spatial locations corresponding to all those time points. This set Priest calls Σt. In the diagram we see how this spatial spread matches the temporal spread, and that only those falling within those spread are true. Another concept Priest uses is ‘degenerate’ which seems to mean that a set of locations corresponds to a single time point and not a set of time points, but please consult the text to be sure, p.178.]

The appropriate state description for the Hegelean account will, of course, be different, incorporating, as it does, the spread hypothesis. In accordance with the hypothesis, there is an interval containing t, θt (which may depend not only on t but also on f) such that, in some sense, if t' ∈ θt, b’s occupation of its location at t' is reproduced at t. I suggest that a plausible formal interpretation of this is that the state description of b at t is just the ‘‘superposition’’ of all the Russellean state descriptions, vt', where t' ∈ θt. More precisely, it is the evaluation, v, given by the conditions

Let us call this the Hegelean state description of the motion. Suppose we write Σt for the spread of all the points occupied at t, i.e., for {f(t') | t' ∈ θt}. If Σt is degenerate, that is if Σt={f(t)}, then the Hegelean state description is identical with the Russellean one. If it is not, then, as may easily be seen, the condition on the righthand side of (2b) is satisfied by all r, and we may depict the Hegelean state description as follows

[178]

[Priest then discusses the contradiction that would arise if Σt were not degenerate, which I think means that there are many locations corresponding to just one time point, but I am not sure. Perhaps the contradiction he describes is that for one time point, the object is in many places, but that means it is both in one certain such place and not in it during the same instant. But suppose that the object remains in one position within Σt. This does not lead to a contradiction and in fact describes a state of rest. It is even compatible with the Russellean description of rest (being at the same place throughout different times). However, even with all points in Σt being the same, there can still be a contradiction. This would happen if the moments surrounding very near the given time point (or θt) extend beyond the scope of , and thus the object will be again both in one location in that temporal spread and not in that location. But, since the temporal spread θt is very brief, “this unstable state of affairs can never last for very long.” He then introduces the idea of the derivative df/dt. It seems like he is saying that since the object does not move far enough to register an assignable value greater than 0, and thus finitely speaking does not change assignable locations, hence making it not a contradiction. But please consult p.179 and check.]

As the picture shows, if Σt is not degenerate, then at t a number of contradictions are realised. For all r ∈ Σt, 1 ∈ vt(Br ∧ ¬Br). Σt may be degenerate for one of two reasons. The first is that θt may itself be degenerate. That is, θt={t}. The other is that, though θt is not degenerate, f is constant over it. Now θt is not, in general, degenerate (or the Hegelean account collapses into the Russellean one). It is quite plausible to suppose that its length depends on the velocity of b, so that the faster b is going the more difficult it is to ‘‘pin it down’’. At any rate, provided θt is non-degenerate, if b satisfies the Russellean conditions of motion at | t (namely that at arbitrarily close points of time it is to be found elsewhere), then contradictions will be realised at t. If, on the other hand, a body occupies the same spot at all times in θt, St will be degenerate and no contradiction will be realised. It is possible (for all I have said so far) for a body to satisfy the Russellean conditions for rest, that is, to occupy the same place over a period of time, and yet for a contradiction to be realised during that time. This will happen at t if θt extends beyond this period of constant position. But since θt is very small (maybe in the order of Planck’s constant?) this unstable state of affairs can never last for very long. We might even suppose that if df /dt=0 then θt is degenerate. Now, if f is constant for a period around t, then df /dt=0 at t. In this case, therefore, no contradiction is realised at t. [178-179]

[Priest finishes by noting that we might need to know whether or not θt extends beyond t or if t is the least upper bound of θt. This is a problem, because it might imply there can be backwards causation. Priest provides a solution. Please see page 179 for details.]

[The following is summary and quotation. All boldface, underlining, and bracketed commentary are mine. Please see the original text, as I did not follow it closely. Proofreading is incomplete, so mistakes are still present.]

Bertrand Russell

Principles of Mathematics

Part 5: Infinity and Continuity

Ch.41: Philosophical Arguments Concerning the Infinitesimal

Brief Summary:

The concept of the infinitesimal as it is used in calculus involves the idea that there is a sequence of consecutive infinitely small values. But were this to be true, then there would be a one-one correspondence between the terms of one series and those of the other to which the first is being differentially related, resulting always in the ratio 1/1. However, calculus finds these differential ratios to have many values other than 1/1. So the infinitesimal leads to contradictions and must not be used in mathematics and presumably for that reason not in philosophy either.

Summary

§315

Previously Russell had argued against using the concept of the infinitesimal to account for continuity. Now he will address philosophical arguments which want to defend the infinitesimal. For this he will examine Cohen’s Princip der Infinitesimalmethode und seine Geschichte.

§316

The differential in calculus no longer needs the concept of the infinitesimal.

In the above exposition, the differential appeared as a philosophically unimportant application of the doctrine of limits. Indeed, but for its traditional importance, it would scarcely have deserved even mention. And we saw that its definition nowhere involves the infinitesimal. The dx and dy of a differential are nothing in themselves, and dy/dx is not a fraction. Hence, in | modern works on the Calculus, the notation f' (x) has replaced dy/dx, since the latter form suggests erroneous notions [344]

But Cohen treats “the dx and the dy treated as separate entities, as real infinitesimals, as the intensively real elements of which the continuum is composed (pp. 14, 28, 144, 147).” [344] Because Cohen does not defend this concept of the infinitesimal, it seems to not be in question. “This view is certainly assumed as self-evident by most philosophers who discuss the Calculus. Let us see for ourselves what kind of grounds can be urged in its favour.” [344]

§317

Although Cohen may have understood the infinitesimals in terms of space and time, Russell will concern himself only with “such arguments as can be derived from purely numerical instances.” [344]

§318

Cohen rejects “the view that the infinitesimal calculus can be independently derived by mathematics from the method of limits.” It seems his reasoning is that (a) the method of limits presupposes a conception of equality [which is problematic for some reason, perhaps because it presupposes the idea of magnitude, but that is the second problem], and (b) the method of limits presupposes the concept of magnitude, but the concept of magnitude presupposes the concept of limit. [It may not be necessary now to fully understand how Cohen arrive at these conclusions. Russell will just show why they do not hold in mathematics.]

This method, he says (p. 1), “consists in the notion that the elementary conception of equality must be completed by the exact notion of the limit. Thus in the first place the conception of equality is presupposed. . . . Again, in the second place, the method of | limits presupposes the conception of magnitude. . . . But in the presupposed conception of magnitude the limiting magnitude is at the same time presupposed. The equality which is defined in the elementary doctrine of magnitude pays no attention to these limiting magnitudes. For it, magnitudes count as equal if and although their difference consists in a limiting magnitude. Hence the elementary conception of equality must be—this is the notion of the method of limits—not so much completed as corrected by the exact conception of the limit. Equality is to be regarded as an earlier stage of the limiting relation.” [Russell 344-345, quoting Cohen p.1]

§319

But, Russell notes, “equality has no relevance to limits”. [Russell’s explanation for this can be found pp.345-346. He notes that the simplest concept of limit is ω, the limit of the ordinal numbers, but it does not involve the concept of equality (perhaps because it does not equal the largest of the ordinals; it is a figure above it). He gives the example of a diminishing series tending toward a value, which might seem like the sum equals the limit value, but in fact we do not need to think of it that way. See the noted pages for details.]

And Russell has already explained how magnitude is not involved in the concept of limits [limits are understood in terms of numerical series and not as series of diminishing magnitudes. See p.346 for details].

§320

[Russell then addresses the argument that magnitude presupposes the concept of limits, but limits presuppose the concept of magnitude. Yet, limits do not require the concept of magnitude. See p.346 for details.]

§321

The biggest mistake Cohen makes is that he thinks limits introduce a new meaning of equality. For magnitudes, there is only one meaning, and it does not involve the notion of approximation [no matter how close]. Cohen thinks that numbers do not have equality but only identity, which is (misleadingly) expressed using the equals sign. [Perhaps magnitudes would be two things that have identical values, and thus can be equal like two weights on a scale. But if two numbers are equal, that means they are the same number. There is not two cases of 2. There is only 2, but 2 can be expressed many ways, like 8/4. What this has to do with calculus is not so clear. But Russell goes on to say it seems that for Cohen, the infinitesimal magnitude added to a value y is equal to y because it is so close an approximation. Russell then reminds us that there is no such thing as dx and dy in calculus. Please read the text to get a more detailed understanding of Russell’s argument, pp.346-347.]

I imagine that what Cohen means may be expressed as follows. In forming a differential coefficient, we consider two numbers x and x + dx, and two others y and y + dy. In elementary Arithmetic, x and x + dx would count as equal, but not in the Calculus. There are, in fact, two ways of defining equality. Two terms may be said to be equal when their ratio is unity, or when their difference is zero. But when we allow real infinitesimals dx, x and x + dx will have the ratio unity, but will not have zero for their difference, since dx is different from absolute zero. This view, which I suggest as equivalent to Cohen’s, depends upon a misunderstanding of limits and the Calculus. There are in the Calculus no such magnitudes as dx and dy. There are finite differences Δx and Δy, but no view, however elementary, will make x equal to x + Δx. There are ratios of finite differences, Δy/Δx, and in cases where the derivative of y exists, there is one real number to which Δy/Δx can be made to approach as near as we like by diminishing Δx and Δy. This single real number we choose to denote by dy/dx; but it is not a fraction, and dx and dy are nothing but typographical parts of one symbol. There is no correction whatever of the notion of equality by the doctrine of limits; the only new element introduced is the consideration of infinite classes of terms chosen out of a series. [347]

§322

[In the following Russell notes another of Cohen’s claims, but that claim is not explained. Perhaps it is saying that because dx and dy are infinitesimal, they do not extend in space, and thus they are inextensive. The extensive can be contrasted with the intensive. In Kant, the extensive magnitude is divisible into metrical parts, but the intensive is not. In ch.21, Russell discusses magnitudes and their measure, and he says that extensive magnitudes are numerically measurable but intensive magnitudes are not; they only admit of more or less. Also in that chapter Russell notes how for Kant, intensive magnitudes are realities that can be more or less in magnitude, like more or less bright. Let’s first look at Russell’s passage in this chapter.]

As regards the nature of the infinitesimal, we are told (p. 15) that the differential, or the inextensive, is to be identified with the intensive, and the differential is regarded as the embodiment of Kant’s category of reality. [347]

[The differential ratio of dx to dy is inextensive for Cohen, because dx and dy are infinitesimal and thus do not extend in space. Being inextensive does not necessarily mean intensive yet. dx and dy are inextensive, as is their ratio. However, their ratio forms a quantity that is intensive, meaning that (using Russell’s definition) they are measurable in terms of being more or less but not numerically by counting parts. Kant’s concept of reality regards it as something that admits of degrees, and thus somehow the differential is the embodiment of Kant’s category of reality. I have not read Cohen yet, so I do not know his reasoning. But to give our own, we might say that Kant (especially according to a Deleuzean interpretation) regards our experiences as being experiences of reality, but these experiences are the experiences of the variations from moment to moment. We only experience degrees of difference and thus only intensities. It is only by means of recollection that we experience by means of synthesis stretches of time and the extensity of spatial objects. Thus reality in its most basic form are variations, which are understood as correlated infinitesimal differences, tiny changes over tiny moments.] [So Cohen understands dx and dy as being terms in a series, or as differences between consecutive terms. Russell will explain why they are neither, and instead they only represent stretches (series of intermediate terms) containing an infinity of terms, or “distances corresponding to such stretches”. So for Russell, dx and dy are not infinitesimals but rather merely tiny finite values that are infinitely divisible like any other finite value. Then Russell distinguishes series of numbers from series of measurable stretches or distances. Space and time, for example, are this second kind that are made of stretches or distances. But, dx and dy are not consecutive terms, because our series is compact (between any two there is another, and as we saw, there cannot be consecutive terms in a compact series, because there is no ‘next’ term; for there always is a ‘more next’ term, then another, and another, without end.) After considering some complications, Russell finds a possible way to tentatively conceive of dx and dy as being the distances of consecutive points. Russell will show why this is still absurd. He thinks it leads to the conclusion that all differential relations dx/dy would have to have the same value, either positive or negative 1. He could perhaps be saying the following. Suppose like Cohen we think that although a distance is infinitely divisible, it ultimately divides into smallest parts. These parts do not have a finite value. However, there are infinitely many such parts in a finite distance. dx and dy are thought of as such infinitesimal distances. Thus any finite distance along the x axis (or x series) is made up of an infinity of dx’s, which measure the distances between the consecutive points, and likewise for y. But the points between which dx and dy stand correlate in a one-one fashion, since these points are real values of the number line. This would seem to imply that dx and dy are always constant values, and thus dx/dy is always positive or negative one. This is because no other points intervene between them. Thus regardless of the supposed relative values of any dx/dy pairing, each themselves cannot have a value any different than the equally spaced points on which they are found, and thus must always be equal. Most likely Russell is making a different argument, which I cannot discern, so it is important to read the ‘mathematical arguments’ on page 348 for a more certain interpretation. Russell then puts these mathematical arguments aside, and says that since dx/dy have a numerical ratio, they must be numerically measurable, even though they are intensive magnitudes. (And recall that for Russell intensive magnitudes are not numerically measurable.) But Russell does not see how we might numerically measure them. So first we suppose that x and y are numbers. Then we suppose that x and x + dx are consecutive. Now, how are we to regard y + dy? We have four options. They either (a) are consecutive, (b) are identical, (c) have a finite number of terms between them, or (d) have an infinite number of terms between them. Cases c and d I think would be cases where dy is a stretch, that is, a series of terms between two end terms. Russell says that if it is a stretch, then dy/dx will always be either zero, integral, or infinite. I do not know why. Let’s suppose that these results follow from b, c, and d, as a possible way to start our explanation. If y and y + dy are identical, that means dy is 0, and making dy / dx be 0 over some other figure and thus 0. If there is a finite number of terms between y and y + dy, then that means dy/dx would have some integer value, and maybe that is what Russell means by ‘integral’. However, I do not know what he means here; if it has something to do with integral calculus I cannot discern it; and that dy/dx would have an integer value does not to me seem problematic, so I cannot interpret that. If there are an infinity of terms between y and y + dy, that means we have infinity over one (or some finite value) and thus dy/dx is infinity. In all three cases this is absurd (although the absurdity of the second case I cannot understand. It is also possible that the results “zero, or integral, or infinite” are not results of cases b, c, and d respectively. But if that were so, I understand the situation even less.) Russell then goes on to say that even if y is not constant, dy/dx must be positive or negative 1. It seems he proves this by considering the two ways we can conceive of dy and dx, that is, as being either stretches or distances. If they were stretches, that means no matter the size of y, it will have the same size of infinity of dy-components as x has of dx-components, and they will correlate always in a one-one fashion. Since for stretches the number of terms determines the magnitude, and because the number of terms is equal and correspondent in a one-one fashion in both x and y, then dy/dx will always be 1/1. He then says that if y is not constant (sticking still with dy and dx being stretches), dy/dx will still have to be positive or negative one. He has us consider the function y = x2. And here x and y are positive real numbers. But again, if they are stretches, the same one-one correspondence will apply and thus the same problem results. Now he has us consider if we measure by distances and not stretches. He seems to be using the same reasoning. He says dy and dx are always the distance from one number to the next along the distances y and x. He then has us consider a function for which dy/dx = 2 for x = 1 and y = 1. On the one hand, the function tells us dy/dx should be two. But since there is this one-one correspondence and since x and y are equal, it would also have to be 1/1, which is absurd. This means that no matter how we conceive of consecutive values, it will lead to absurdities when applied in calculus. I would like to point out possible reasons we might not have to come to Russell’s conclusion. Russell’s argument begins by conceptualizing the infinitesimals as being the consecutive intervals between the real number values taken to be infinitely close. So under this conception, we would think that there are an infinity of infinitely small increments making up the value x and the value y, and each such increment corresponds to a real number value, which is like the total of all the infinitely small increments leading up to it. Let’s think about a geometrical interpretation, for example the curve described by y = x2. When x is 2, y is 4, and the rise/run of the tangent at that place along the curve is 4/1. Russell’s problem is that this implies that as we move to the next real number value, x + dx, we would skip over 3 points of y on the y axis, since we are jumping by 4. (In fact, since the variations are exponential, when we get to the next one, we might have jumped over even more than 4.) But that does not mean the next four real number values do not have a corresponding y value. Consider: the function says for example we are going from (2,4) to (3,9) and so on [for (x,y) coordination, with the difference between values assumed to be infinitely small]. Thus we see that we are always skipping y values when we are keeping the x values constant. However, what happens when we go up the scale of y values? What if we went from (2,4) to (x,5)? What is the value of x? Would it not also have a value, which would be between the afore-determined x-values? And as we go up the scale of y, the x values will grow relatively slower. There seems then to be an impossible contradiction if we assume consecutive infinitesimal values making up the spaces between points on the x and y axes. But perhaps there is a flaw in how Russell sets up the problem. He equates real numbers along x and y with infinitesimal increments along x and y (or along x/y). When two successive numbers are real, then there is always another between them. So there are not consecutive real numbers whose values can be assigned. The infinitesimal interval would be smaller than the interval between any givable pair of reals. So maybe we cannot, like Russell does, equate the infinitesimal increments with the real’s increments. So if we go up an infinitesimal increment along the x axis, that does not mean that it must correspond to an increment along the y which is equal in magnitude. How we are to better conceptualize such successions of infinitesimals I am not sure, but it does seem to be fairly certain that they are not equal to the succession of real numbers and thus Russell’s criticism might not hold. I have placed the entirety of this paragraph below, because it deserves a better interpretation than I can give it.]

As regards the nature of the infinitesimal, we are told (p. 15) that the differential, or the inextensive, is to be identified with the intensive, and the differential is regarded as the embodiment of Kant’s category of reality. This view (in so far as it is independent of Kant) is quoted with approval from Leibniz; but to me, I must confess, it seems destitute of all justification. It is to be observed that dx and dy, if we allow that they are entities at all, are not to be identified with single terms of our series, nor yet with differences between consecutive terms, but must be always stretches containing an infinite number of terms, or distances corresponding to such stretches. Here a distinction must be made between series of numbers and series in which we have only measurable distances or stretches. The latter is the case of space and time. Here dx and dy are not points or instants, which alone would be truly inextensive; they are primarily numbers, and hence must correspond to infinitesimal stretches or distances—for it would be preposterous to assign a numerical ratio to two points, or—as in the case of | velocity—to a point and an instant. But dx and dy cannot represent the distances of consecutive points, nor yet the stretch formed by two consecutive points. Against this we have, in the first place, the general ground that our series must be regarded as compact, which precludes the idea of consecutive terms. To evade this, if we are dealing with a series in which there are only stretches, not distances, would be impossible: for to say that there are always an infinite number of intermediate points except when the stretch consists of a finite number of terms would be a mere tautology. But when there is distance, it might be said that the distance of two terms may be finite or infinitesimal, and that, as regards infinitesimal distances, the stretch is not compact, but consists of a finite number of terms. This being allowed for the moment, our dx and dy may be made to be the distances of consecutive points, or else the stretches composed of consecutive points. But now the distance of consecutive points, supposing for example that both are on one straight line, would seem to be a constant, which would give dy/dx = ±1. We cannot suppose, in cases where x and y are both continuous, and the function y is one-valued, as the Calculus requires, that x and x + dx are consecutive, but not y and y + dy; for every value of y will be correlated with one and only one value of x, and vice versâ; thus y cannot skip any supposed intermediate values between y and y + dy. Hence, given the values of x and y, even supposing the distances of consecutive terms to differ from place to place, the value of dy/dx will be determinate; and any other function y' which, for some value of x, is equal to y, will, for that value, have an equal derivative, which is an absurd conclusion. And leaving these mathematical arguments, it is evident, from the fact that dy and dx are to have a numerical ratio, that if they be intensive magnitudes, as is suggested, they must be numerically measurable ones: but how this measurement is effected, it is certainly not easy to see. This point may be made clearer by confining ourselves to the fundamental case in which both x and y are numbers. If we regard x and x + dx as consecutive, we must suppose either that y and y + dy are consecutive, or that they are identical, or that there are a finite number of terms between them, or that there are an infinite number. If we take stretches to measure dx and dy, it will follow that dy/dx must be always zero, or integral, or infinite, which is absurd. It will even follow that, if y is not constant, dy/dx must be ±1. Take for example y = x2, where x and y are positive real numbers. As x passes from one number to the next, y must do so likewise; for to every value of y corresponds one of x, and y grows as x grows. Hence if y skipped the number next to any one of its values, it could never come back to pick it up; but we know that every real number is among the values of y. Hence y and y + dy must be consecutive, and dy/dx = 1. If we measure by distances, not stretches, the distance dy must be fixed when y is given, and the distance dx when x is given. Now if x = 1, y = 1, dy/dx = 2; but, since x and y are the same number, dx and dy must be equal, since | each is the distance to the next number: therefore dy/dx = 1, which is absurd. Similarly, if we take for y a decreasing function, we shall find dy/dx = − 1. Hence the admission of consecutive numbers is fatal to the Calculus; and since the Calculus must be maintained, the Calculus is fatal to consecutive numbers. [347-349]

§323

[First Russell notes that perhaps some of the problems that have arisen result from the conceptualization of going from one term to the next being a matter of physical motion (like a point moving along the x-axis) when it is really more of a numerical progress without real temporal and spatial properties. He then goes on to challenge Cohen’s idea that inextensive infinitesimals are equatable with intensive magnitudes. To explain his reasoning for this, let’s consider first an example of an intensive magnitude, let’s say the brightness of a light. We would never say that it is smaller than some extensive magnitude. It is just a different kind of magnitude. But the infinitesimal is smaller than any extensive magnitude, and for that reason should not be considered intensive.]

§324

[In this last paragraph, Russell sums up his argument so far against infinitesimals: they are (1) unnecessary (because they are not needed for calculus), (2) erroneous (because he showed in a prior chapter that they are obtained through an “illegitimate use of mathematical inductions) and (3) self-contradictory (because they lead to such contradictions as the one mentioned above regarding their consecutivity).]

We cannot, then, agree with the following summary of Cohen’s theory (p. 28): “That I may be able to posit an element in and for itself, is the desideratum, to which corresponds the instrument of thought reality. This instrument of thought must first be set up, in order to be able to enter into that combination with intuition, with the consciousness of being given, which is completed in the principle of intensive magnitude. This presupposition of intensive reality is latent in all principles, and must therefore be made independent. This presupposition is the meaning of reality and the secret of the concept of the differential.” What we can agree to, and what, I believe, confusedly underlies the above statement, is, that every continuum must consist of elements or terms; but these, as we have just seen, will not fulfil the function of the dx and dy which occur in old-fashioned accounts of the Calculus. Nor can we agree that “this finite” (i.e. that which is the object of physical science) “can be thought as a sum of those infinitesimal intensive realities, as a definite integral” (p. 144). The | definite integral is not a sum of elements of a continuum, although there are such elements: for example, the length of a curve, as obtained by integration, is not the sum of its points, but strictly and only the limit of the lengths of inscribed polygons. The only sense which can be given to the sum of the points of the curve is the logical class to which they all belong, i.e. the curve itself, not its length. All lengths are magnitudes of divisibility of stretches, and all stretches consist of an infinite number of points; and any two terminated stretches have a finite ratio to each other. There is no such thing as an infinitesimal stretch; if there were, it would not be an element of the continuum; the Calculus does not require it, and to suppose its existence leads to contradictions. And as for the notion that in every series there must be consecutive terms, that was shown, in the last chapter of Part III, to involve an illegitimate use of mathematical induction. Hence infinitesimals as explaining continuity must be regarded as unnecessary, erroneous and self-contradictory. [350]